Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). 3 Let A 14). Compute [AB (2pt) Enter your answer here and T(A) C (2pt). Enter your answer here
Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Show that T is linear Q1 17 Points Let b T: M2x2(R) + P2 (R), H (2a+b)x2 + (b – c)x+(c – 3d). с d Let 1 0 0 0 B = (( b); C8 1 0 0 0 0 1 :)C. 11), 1) (7.1)) i), (6 ;)) 1 0 1 B = (CO 2 -1 1 1 1 C = (x², x, 1) C' = (x + 2, x + 3, x2 – 2x – You may assume that...
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
Let T. M2(R) →P2(R) be defined by T.(Iga)-(+b) + (b+c) Let T2: P2 (R) → Pl (R) be defined by Tap(x))-p' (x) (c+ d)x2 2. Find Ker(T2 . T) and find a basis for Ker(T2。T).
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
Let a(x) = 1/x, b(x) = x,c(x) = 2x + 6, and d(x) = x2. Express each of the following functions as a composition of two of the given functions. (a) f(x) = 2x + 6 f(x) = (bo c)(x) f(x) = (co b)(x) R(x) = (bo a)(x) f(x) = (doc)(x) f(x) = (a o b)(x) F(x) = (co d)(x) (b) 9(X) - 1/X2 9(x) = (doc)(x) 9(x) = (b o a)(x) g(x) = ( ac)(x) 9(x) = (a o...
Consider a subset alpha={x+x2,1+x2,1 2x+2x2}ofP2(R). (a) Show that alpha is a basis for P2(R). (b) For f(x) = 1 + x + x2 2 P2(R), find its coordinator vector [f] alpha with respect to alpha. (c) Let = {1, x, x2} be the standard basis for P2(R), and let f(x) = a + bx + cx2 and g(x) = p+qx+rx2 be the elements of P2(R) and k 2 R. Prove that [f+g] = [f] +[g] and [kf] = k[f] and...
4. (10 points) Let T:P2 + M2x2(R) be the linear map Tax+bæ + c) =( and 1/1 0 1 B' = 0 1 0 0 O 0) (10) be the standard basis for the codom(T). Using 1 any basis B for P2, compute TB.